# TEXAS A&amp;M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 251-508

### Final Exam version A, Dec 2008

On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.

Name (print):

In all questions, no analytical work — no points .

1.

Find an equation for the line passing through the point A (3 , 2 , 0) and orthogonal to the line

 x = 1 + t,

 y = 2 − t,

 z = 1

2.

Find the absolute minimum and maximum values of f ( x, y ) = 2 x 2 + x + y 2 − 2 on the set

D = { ( x, y ) | x 2 + y 2 ≤ 4 } .

3.

Evaluate the integral by changing to cylindrical coordinates

Z

1

1

Z

1

− x 2

1

− x 2

Z

2

− x 2

− y 2

( x 2 x 2

+ y 2

+ y 2 ) 3 / 2 dz dy dx

4.

Evaluate

Z

C

(2 x sin y ) dx + ( x 2 cos y − 3 y 2 ) dy, where C is the path from (0 , 0) to (1 , 1) along the parabola y = x 2 .

5.

Verify the identities

1.

∇ r = r /r ,

2.

∇ · r = 3,

3.

∇ × r = 0 , where r = h x, y, z i , r = | r | and ∇ =

D

∂x

, ∂

∂y

, ∂

∂z

E

.

6.

Evaluate

Z

C x 2 z dx + xy 2 dy + z 2 dz, where C is the intersection of the plane x + y + z = 1 and the cylinder x 2 + y 2 = 9 oriented counterclockwise as viewed from above. Hint: Use Stokes’ Theorem.

7.

Bonus question +10% (no partial credit): Since Krokodil Gena is still doing the dishes for cheating in the calculus riddles game in spring semester (see the last semester’s final exam), he refuses to help Cheburashka with his homework. You are now the only one who can help poor Cheburashka (and his professor is ok with that, good that you asked).

He has to find the formulas for the positive numbers x , y and z whose sum is 100 and such that the product x a y b z c is maximal.

Points: /30